I’m not sure how to check the ECU exactly however, those engines became more and more efficient in the New Years. That’s part of why you see the change from wires to coils in the newer engines (after 2001 or 2002). If you can figure out what year the new engine (it’s probably an 02 or 03 es300 if it has coils and not wires) is from you can cross reference the part number for that ECU with the ECU you are currently using to make sure they’re the same. If they aren’t the problem could be with the ECU.

if the ECU Isn’t right for the car you can see issues like it spraying too much fuel, backfiring, and misfiring badly even with good coils so if you find one at a junkyard or cheap nearby I would invest and put in the proper ECU.
another simple fix is to make sure the coils are not leading to the wrong places, make sure they aren’t on the wrong cylinders.

 

My engine was a 95 and it had coils, Also a 02 or 03 engine is not compatible with a 95 es300 because it has variable valve timing and a 95 model did not have that...

 

Don't forget about the injectors they should also be tested to see if they are getting a signal. Are you sure there are no pinched wires, ground wires that are not connected? Vacuum lines missing or loose? These are the most common things I see after an engine swap.

 

Dont see any pinched wires or vacuum lines that are loose or missing. What is interesting though when the mechanic swapped my motor out he said there were two black wires that were coming from the wire harness that were not connected to anything. When he swapped to motor in he grounded those 2 wires. Mind you before the engine swap I had none of these issues and no electrical problems I just decided to swap the motor because a rod started to knock...

 

He grounded wires without knowing what they are for? That's some hackery right there.

 

So after ungrounding the 2 wires and installing new coils on the back 3 cylinders the 0301,0303 and 0305 codes are gone but now there is a p1300 code...

 

p1300 is just a multiple cylinder misfire so as long as you can visibly see and hear that there’s no misfire just disconnect the battery and reconnect it. The code is probably from the previous misfire and the computer hasn’t cleared it yet

 

There is still a miss at idle, and after clearing the p1300 with the scanner after revving the engine up in rpms the same p1300 code appears again...

 

The injectors and coils need to be back probed and monitored with a scan tool/scope otherwise you'll be endlessly throwing parts at the problem.